Can anyone work out the How Much the Red Color Area?

Jul 2019
wenzhou china
See this drawing:
Square side = 10,
how much is the red area?
Aug 2006
See this drawing:
Square side = 10,
how much is the red area?
View attachment 39464
The entire coloured area inside the square is $100$ square units.
To answer your question, you must subtract the areas:
1) the brown area,
2) the complete area inside the large complete circle,
3) the three green areas.

Now I will not work this completely for you. You can pay a homework service to do that.
The area inside the large complete circle is $(5)^2\pi$ square units
Can you explain why the brown area is $\frac{1}{4}\left(100-25\pi)\right)$ square units?
How do you find the area of one of the three green areas?
  • Like
Reactions: 1 person
Nov 2012
Normal, IL USA
I'm not the OP, but I'm been looking at this problem and I'm having trouble with it.

I can calculate the red and green areas, but I can only get a decimal approximation. Each of the 6 small red areas measures about 0.728505896. Each of the green areas measures about 3.908034123. I can do this using a method suggested by a post from Christian Blatter that I saw on mathstackexchange. Refer to the diagram below:

red area calculation.png

First we consider that the square is only 2 units on a side, and define a coordinate system at the center. That way the inscribed circle is a unit circle, which simplifies some things. Then it's easy to find the point where the inscribed circle and the quarter circle intersect, and thus to find the areas of the two sectors and the triangle (Mr. Blatter found the area of the triangle using the wedge product, whereas I used Heron's formula; the result is the same either way). Then we just subtract those three area from the trapezoid, of the which the area is 3/2, and multiply the result by 25.

Now, here's my problem. The way the problem was originally stated, as well as the way you answered it, suggest to me that it should be possible to do this using only exclusion and inclusion, and the result should be something of the form \(\displaystyle {q_1} - {q_2}\pi \) where \(\displaystyle {q_1}\) and \(\displaystyle {q_2}\) are both rational numbers. Surely there must be a \(\displaystyle \pi \) in there someplace!

I did discover a couple of things using inclusion and exclusion beyond what you mentioned in your post, things you no doubt already know, but I'm spell them out just so you know that I know them too. First, there's a 4 to 1 proportionality running through the diagram. For instance, in the upper left corner there is an area outside one of the four large quarter circles that is 4 times the size of the brown area, \(\displaystyle 100 - 25\pi \) Also, there are two lens-shaped areas where two of those large quadrants intersect. Each of them has an area of \(\displaystyle 100 - 50\pi \).

I have not been able to calculate the red or green areas based simply on exclusion and inclusion. Apparently, nobody on mathstackexchange has been able to do it either. I showed this problem to a friend of mine who is a retired math professor, and he hasn't been able to do it either.

If there is a way to do it and you know what it is, could you give me a hint? Much appreciated.